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The Diophantine equation A² + nB⁴ = C³ has infinitely many integral solutions A, B, C for any fixed integer n. The case n = 0 is trivial. By using a new polynomial identity we generate these solutions, and then give conditions when the solutions are pairwise co-prime.

Parametrization of integral values of polynomials

Giulio Peruginelli (2010)

Actes des rencontres du CIRM

We will recall a recent result about the classification of those polynomial in one variable with rational coefficients whose image over the integer is equal to the image of an integer coefficients polynomial in possibly many variables. These set is polynomially generated over the integers by a family of polynomials whose denominator is $2$ and they have a symmetry with respect to a particular axis.We will also give a description of the linear factors of the bivariate separated polynomial $f (X) - f (Y)$ over a...

Parametrized solutions of Diophantine equations

Günter Lettl (2004)

Mathematica Slovaca

Parametrizing SL₂(ℤ) and a question of Skolem

Umberto Zannier (2003)

Acta Arithmetica

Partitions and indefinite quadratic forms.

G.E. Andrews, F. Dyson, D. Hickerson (1988)

Inventiones mathematicae

Pell and Pell-Lucas numbers of the form $- 2^{a} - 3^{b} + 5^{c}$

Yunyun Qu, Jiwen Zeng (2020)

Czechoslovak Mathematical Journal

In this paper, we find all Pell and Pell-Lucas numbers written in the form $- 2^{a} - 3^{b} + 5^{c}$ , in nonnegative integers $a$ , $b$ , $c$ , with $0 \leq max {a, b} \leq c$ .

Perfect powers expressible as sums of two fifth or seventh powers

Sander R. Dahmen, Samir Siksek (2014)

Acta Arithmetica

We show that the generalized Fermat equations with signatures (5,5,7), (5,5,19), and (7,7,5) (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the non-existence of non-trivial primitive integer solutions for the signatures (5,5,11), (5,5,13), and (7,7,11). The main ingredients for obtaining our results are descent techniques, the method of Chabauty-Coleman, and the modular approach to Diophantine equations.

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Pairs of additive equations IV. Sextic equations

Pairs of additive equations of small degree

Pairs of additive forms and Artin's conjecture

Pairs of cubic forms in many variables

Pairs of Pell equations having at most one common solution in positive integers.

Palindromic powers.

Parametric solutions for some Diophantine equations.

Parametric Solutions of the Diophantine Equation A² + nB⁴ = C³

Parametrization of integral values of polynomials

Parametrized solutions of Diophantine equations

Parametrizing SL₂(ℤ) and a question of Skolem

Partitions and indefinite quadratic forms.

Pell and Pell-Lucas numbers of the form $- 2^{a} - 3^{b} + 5^{c}$

Perfect powers expressible as sums of two fifth or seventh powers

Perfect powers in arithmetic progression. A note on the inhomogeneous case

Perfect powers in arithmetical progression (II)

Perfect powers in linear recurring sequences

Perfect powers in products of integers from a block of consecutive integers

Perfect powers in products of integers from a block of consecutive integers (II)

Perfect powers in products of terms in an arithmetical progression III

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